A remarkable construction of optimal locally repairable codes via subcodes of Reed-Solomon codes was given by Tamo and Barg in [1]}. There are some constraints on choice of locality due to the existence of good polynomials, and the length is upper bounded by the code alphabet size \(q\) . This construction was generalized via the automorphism group of a rational function field in [2]} and it turns out that there are more flexibility on locality and the code length can be \(q+1\) . Based on the classical MDS conjecture, one should wonder if \(q\) -ary optimal locally repairable codes can have length bigger than \(q+1\) . Surprisingly, several optimal locally repairable codes with length around \(q^2\)  are constructed from algebraic surfaces in [3]}. By making use of automorphism groups of elliptic curves, optimal locally repairable codes with length up to \(q+2\sqrt{q}\)  can be constructed [4]}, [5]}. Algebraic surfaces are employed to construct two families of optimal locally repairable codes [6]}. In these constructions, there are no restriction on the minimum distance of optimal locally repairable codes.
